Second order cumulants: Second order even elements and $R$-diagonal elements
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Bibliographic record
Abstract
We introduce R -diagonal and even operators of second order. We give a formula for the second order free cumulants of the square x^2 of a second order even element in terms of the second order free cumulants of x . Similar formulas are proved for the second order free cumulants of aa^* , when a is a second order R -diagonal operator. We also show that if r is second order R -diagonal and b is second order free from r , then rb is also second order R -diagonal. We present a large number of examples, in particular, the limit distribution of products of Ginibre matrices. We prove the conjectured formula of Dartois and Forrester for the fluctuations moments of the product of two independent complex Wishart matrices and generalize it to any number of factors.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it